4.1.22 · D1Implicit differentiation — technique, applications
Do quantities x y equation se bandhe ho sakte hain — ek ko move karo to doosra bhi move hota hai. Implicit differentiation bas yeh poochna hai: "jab x y d x d y
Yeh page kuch bhi assume nahin karta . Parent note ne jo bhi notation use ki hai, woh saari neeche kholi gayi hai, usi order mein jo tumhe chahiye. Agar parent page par koi symbol magic lagaa, woh yahaan decode ho jaayega.
Ek variable ek slot hai jo alag-alag number values rakh sakta hai. Slots ko hum letters se naam dete hain — usually x choose karte hain, aur y depend karta hai.
Ek number line imagine karo: x x
Ek function ek rule hai jo x y = f ( x ) y f x x f y
Figure dekho: neeche ki line par input dot x f y Yahi picture reason hai ki hum baad mein d x d y — yeh dikhata hai ki y x chained hai, free nahin.
Definition Coordinate point( a , b ) point ( a , b ) a b
Definition Curve of an equationEk equation ka curve un sabhi points ( x , y ) x 2 + y 2 = 25 5
Figure mein notice karo: ek vertical line circle ko do jagah kaatti hai. Yahi visual proof hai ki circle ek function y = f ( x ) nahin hai — function sirf ek y x
Intuition Explicit vs implicit — picture mein
Explicit y = f ( x ) y Implicit F ( x , y ) = 0 x y
(Yahaan F ( x , y ) x y F ( x , y ) = 0 x 2 + y 2 − 25 = 0
Δ x Δ y Δ ek change in ke liye. Δ x x Δ y y Δ x = x new − x old
Curve par do nearby dots imagine karo. Δ x Δ y Ratio Δ x Δ y
Curve par kisi point par tangent line woh straight line hai jo wahan curve ko kiss karti hai — woh ek hi jagah touch karti hai aur, uss jagah ke aas-paas, bilkul wahi direction point karti hai jis taraf curve ja rahi hai. Socho ek ball curve ke andar roll kar rahi hai: woh flat floor jis par woh momentarily rest karegi wahi tangent hai.
Figure mein, amber secant lines dekho: har ek hamare fixed white dot ko aage ek doosre dot se jodti hai. Jaise doosra dot slide karta hai andar (gap Δ x tangent line par settle nahin ho jaati. Woh settling motion hi limit hai, aur uski final steepness hi derivative hai. Dekho Tangent and Normal Lines .
Definition Limit (informal)Limit woh value hai jis par koi quantity homes in karti hai jab hum kuch zero ki taraf squeeze karte hain. Likha jaata hai lim Δ x → 0 Δ x Chain Rule ke prerequisites dekho.)
d x d y Derivative woh slope Δ x Δ y jab hum do dots ko slide hone dete hain jab tak woh touch nahin kar lete:
d x d y = lim Δ x → 0 Δ x Δ y . d y d x Δ y Δ x "is instant mein, x y — aur geometrically yeh section 4 ki tangent line ki steepness hai.
Intuition Yeh tool kyun, sirfΔ x Δ y ek exact spot par pin down karne ka — woh tangent ka slope jo curve ko wahan sirf kiss karta hai.
Definition Notation zoo — sabka matlab "derivative"
d x d y d y d x y ′ y f ′ ( x ) f d x d y y ′
Intuition Power rule sach kyun hai (x 2 x x 2 Δ x x ⋅ Δ x Δ x 2 2 x Δ x Δ x 2 x Δ x = 2 x Woh do strips literally hi 2 x Yahi logic cube ke saath teen slabs → 3 x 2
Intuition Constant rule sach kyun haiEk constant ek flat horizontal line hai — kisi bhi run ke liye koi rise nahin. Flat line ki steepness 0 d x d ( 25 ) = 0
Intuition Chain rule sach kyun hai (gear picture)Do gears ek chain mein imagine karo: x y y g y x d x d y g y g ′ ( y ) g x g ′ ( y ) ⋅ d x d y multiply ho jaate hain. Woh product hai woh toll jo har y
Intuition Product rule sach kyun hai (rectangle picture)Ek rectangle socho jiska width u v uv u ′ Δ x v ′ Δ x v ⋅ u ′ Δ x u ⋅ v ′ Δ x u ′ v + u v ′ har side se badalne par ek strip.
Common mistake "Inside rate 1 par badal raha hai, to fuss kyun?"d x d ( x 2 ) x 1 × 1 d x d ( y 2 ) y d x d y nahin hai 1
F ( x , y ) F ( x , y ) donoN x y F ( x , y ) = x 2 + y 2 − 25 F = 0
F x = ∂ x ∂ F F x F y frozen constant maanke.F y = ∂ y ∂ F x ∂
Intuition "Variable ko freeze karna" kaisa dikhta haiF ( x , y ) landscape socho: flat ( x , y ) F F x due east chalo (sirf x y F y due north chalo (x
F = x 2 + y 2 − 25 y F x = 2 x x F y = 2 y d x d y = − F y F x = − 2 y 2 x = − y x
Definition Woh hypothesis jo ise legal banata hai —F y = 0 d x d y = − F y F x F y F y = 0 Implicit Function Theorem exactly yeh kehta hai: jahan bhi F y = 0 , tangled equation F ( x , y ) = 0 y x F y = 0
Intuition Division by zero ka geometric meaning hota haiCircle ka slope d x d y = − y x y = 0 far left aur right points ( ± 5 , 0 ) y = 0 galti nahin hai; curve tumhe bata raha hai ki wahan uski tangent vertical hai (seedha upar-neeche), jiska koi finite slope nahin hota. Vertical line zero horizontal run ke liye infinitely rise karti hai.
Definition Vertical tangentVertical tangent woh point hai jahan curve momentarily seedha upar/neeche ja rahi hoti hai, isliye d x d y F y = 0 d y d x = − F x F y finite hai.
d x d y = 0 − 5 Galat reaction: "Maine galti ki, division by zero!"
Sahi reaction: woh answer information hai — us point par tangent vertical hai. Use failure ke taur par nahin, vertical tangent ke taur par report karo. Confirm karne ke liye check karo ki wahan F y = 0
sin , cos , tan d x d sin x = cos x d x d cos x = − sin x sin ( x y ) = x + y
arcsin arcsin x sin kaun sa angle hai jiska sine x y = arcsin x sin y = x sin − 1 1 x ∈ [ − 1 , 1 ] . Aur kyunki infinitely many angles ek sine share karte hain, hum answer ko ek branch tak fence karte hain: y ∈ [ − 2 π , 2 π ] cos y ≥ 0 cos y = + 1 − x 2 d x d arcsin x = 1 − x 2 1 . Derivatives of Inverse Functions .
ln ln x kis power par special number e ≈ 2.718 x ln ( a ⋅ b ) = ln a + ln b ln ( a b ) = b ln a sums mein turn karta hai. Isliye dono sides log karne se x x Logarithmic Differentiation mein tame hota hai. Iski derivative yeh clean formula hai:
d x d ln x = x 1 , y x d x d ln y = y 1 ⋅ d x d y x x
Neeche ka figure wohi dependency map teen tiers deep mein draw karta hai: derivative build karo (top), chaar rules + partials (middle), payoff technique aur uske uses (bottom). Tier 1 load-bearing hai — chain rule hatao aur neeche sab collapse ho jaata hai.
Jo readers raw graph prefer karte hain, unke liye wohi structure text form mein:
change delta x and delta y
tangent line kisses the curve
implicit equation F equals 0
partials Fx Fy with Fy not zero
tangents inverses log diff related rates
Apne aap test karo — right side cover karo aur zabar se jawaab do.
y = f ( x ) Input x f y
Circle explicit function kyun nahin hai? Ek vertical line use do jagah hit karti hai — ek x y
Δ x x x new − x old
Tangent line kya hai, ek phrase mein? Woh straight line jo curve ko ek point par sirf kiss karti hai, wahan uski direction match karte hue.
d x d y Jab x y
Plain ratio ki jagah limit kyun lete hain? Bending curve par ek exact point ki steepness measure karne ke liye, gap ke across average nahin.
Power rule d x d x 2 = 2 x Side x x Δ x 2 x
Product rule kisi picture mein sach kyun hai? Rectangle uv u ′ v + u v ′
g ( y ) y = y ( x ) g ′ ( y ) ⋅ d x d y
F x Landscape F y
d x d y = − F y F x F y = 0
Circle par ( 5 , 0 ) d x d y = 0 − 5 Ek vertical tangent — slope undefined/infinite hai, koi error nahin.
arcsin x x ∈ [ − 1 , 1 ] y ∈ [ − 2 π , 2 π ] cos y ≥ 0
d x d arcsin x d x d ln x x 1
x x ln ln ln ( a b ) = b ln a
Chain Rule — woh toll-collecting engine jis se har d x d y Product Rule — mixed x y Derivatives of Inverse Functions — apne fenced range par implicit form sin y = x Logarithmic Differentiation — kyun ln Implicit Function Theorem — guarantee (zaroori hai F y = 0 y ( x ) Tangent and Normal Lines — jahan slope d x d y Related Rates — wohi chain-rule logic, time mein differentiate karte hue. 
